Controlling animation speed for different elements
OK I found myself a solution to this problem!

Everything is OK, it is just necessary to edit some properties under Animation window (Screen Layout). 

Steps:


Edit animation.
Under ''Dope Sheet'' press T and change ''Interpolation Mode'' to ''Linear''.
Under ''Dope Sheet'' press T again and change ''Easing (by strenght)'' to ''Sinusoidal''.
Select all objects that are participating in animation and under ''Graph editor'' press Ctrl+E and change ''Easing Type'' to ''Ease In and Out''.


It works perfectly for my need and I hope it helped someone get the answer of how to move objects with same speed, but that it still looks natural.

Cheers!